The conformal loop ensembles (\(\mathrm {CLE}_\kappa \) , \(\kappa \in (8/3,8)\) ) are the canonical conformally invariant probability measures on locally finite collections of non-crossing loops in a simply connected domain \(D \subseteq \mathbf {C}\)  [1]}, [2]}. It is the loop version of the Schramm-Loewner evolution (\(\mathrm {SLE}_\kappa \) ) [3]}. Just like \(\mathrm {SLE}_\kappa \) , \(\mathrm {CLE}_\kappa \)  arises in the context of two-dimensional statistical mechanics as the scaling limit of the collection of all interfaces of certain models at criticality [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. The \(\mathrm {CLE}_\kappa \) 's have the same phases as the \(\mathrm {SLE}_\kappa \) 's [10]} in that when \(\kappa \in (8/3,4]\)  the loops are a.s. simple, disjoint, and do not intersect the domain boundary and if \(\kappa \in (4,8)\)  they are a.s. self-intersecting, intersect each other and the domain boundary. The set of points which are not surrounded by a loop in the \(\mathrm {CLE}_\kappa \)  is called the \(\mathrm {CLE}_\kappa \)  carpet if \(\kappa \in (8/3,4]\)  and the \(\mathrm {CLE}_\kappa \)  gasket if \(\kappa \in (4,8)\) . The reason for this terminology is that in the former (resp. latter) phase a \(\mathrm {CLE}_\kappa \)  is a random analog of the Sierpinski carpet (resp. gasket). It is proven (see [11]}, [12]}, [13]}) that the dimension of the \(\mathrm {CLE}_\kappa \)  carpet or gasket is a.s.
\(d(\kappa ) = 2-\frac{(8-\kappa )(3\kappa -8)}{32\kappa } = 1+\frac{2}{\kappa } + \frac{3\kappa }{32} \quad \text{for}\quad \kappa \in (8/3,8).\) 
